Adaptive kernel estimation for enhanced filtering and pattern classification of magnetic resonance imaging: novel techniques for evaluating the biomechanics and pathologic conditions of the lumbar spine

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University of Tennessee, Knoxville

Trace: Tennessee Research and Creative

Exchange

Doctoral Dissertations Graduate School

5-2016

Adaptive kernel estimation for enhanced filtering

and pattern classification of magnetic resonance

imaging: novel techniques for evaluating the

biomechanics and pathologic conditions of the

lumbar spine

Nicholas Vincent Battaglia

University of Tennessee - Knoxville, [email protected]

Recommended Citation

Battaglia, Nicholas Vincent, "Adaptive kernel estimation for enhanced filtering and pattern classification of magnetic resonance imaging: novel techniques for evaluating the biomechanics and pathologic conditions of the lumbar spine. " PhD diss., University of Tennessee, 2016.

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To the Graduate Council:

I am submitting herewith a dissertation written by Nicholas Vincent Battaglia entitled "Adaptive kernel estimation for enhanced filtering and pattern classification of magnetic resonance imaging: novel techniques for evaluating the biomechanics and pathologic conditions of the lumbar spine." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Biomedical Engineering.

Mohamed R. Mahfouz, Major Professor We have read this dissertation and recommend its acceptance:

Richard D. Komistek, Adrija Sharma, Aly E. Fathy

Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.)

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Adaptive kernel estimation for enhanced filtering and pattern classification of magnetic resonance imaging: novel techniques for evaluating the biomechanics and pathologic conditions

of the lumbar spine

A Dissertation Presented for the Doctor of Philosophy

Degree

The University of Tennessee, Knoxville

Nicholas Vincent Battaglia May 2016

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Acknowledgments

First and foremost I would like to express my gratitude to my graduate advisor Dr. Mohamed R. Mahfouz, whose experience, knowledge, and keen insights have proven invaluable to the success of my graduate career. His passion to science and relentless pursuit of novel frontiers are qualities I hope to emulate as I move forward. In addition, his belief in me as a student and fellow scientist has been instrumental in motivating me to accomplish my goals.

I am also grateful to Dr. Richard D. Komistek, Dr. Adrija Sharma, and Dr. Aly Fathy for serving on my committee and providing me with valuable feedback that aided in the completion of this dissertation.

I would also like to extend my appreciation to my fellow colleagues at the Center for Musculoskeletal Research (CMR) who – no matter how inane, mundane, or insane my commentary – always greeted me with respect and professionalism. I appreciate all of the hard

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work that Rebecca Robertson contributed to helping me function efficiently within the CMR. I am greatly indebted to Lyndsay Bowers for going above and beyond any obligations or expectations of her to help me whenever possible.

I am eternally grateful for the unwavering support I have received from my immediate family. To my parents, Michelle and Daniel, for the guidance and emotional support; my brother, Andrew, for instilling courage and perseverance; my sister, Kelly, for the insight and wisdom (both bestowed and taught); and brother-in-law, Joel, for lessons in curiosity.

Finally, I could not have made this journey without the support and drive of my wife Aimee and our two children, Jack and Eli. I love you.

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Abstract

This dissertation investigates the contribution the lumbar spine musculature has on etiological and pathogenic characteristics of low back pain and lumbar spondylosis. This endeavor necessarily required a two-step process: 1) design of an accurate post-processing method for extracting relevant information via magnetic resonance images and 2) determine pathological trends by elucidating high-dimensional datasets through multivariate pattern classification. The lumbar musculature was initially evaluated by post-processing and segmentation of magnetic resonance (MR) images of the lumbar spine, which characteristically suffer from nonlinear corruption of the signal intensity. This so called intensity inhomogeneity degrades the efficacy of traditional intensity-based segmentation algorithms. Proposed in this dissertation is a solution for filtering individual MR images by extracting a map of the underlying intensity inhomogeneity to adaptively generate local estimates of the kernel’s optimal bandwidth. The adaptive kernel is implemented and tested within the structure of the non-local means filter, but also generalized and extended to the Gaussian and anisotropic diffusion filters. Testing of the proposed filters

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showed that the adaptive kernel significantly outperformed their non-adaptive counterparts. A variety of performance metrics were utilized to measure either fine feature preservation or accuracy of post-processed segmentation. Based on these metrics the adaptive filters proposed in this dissertation significantly outperformed the non-adaptive versions. Using the proposed filter, the MR data was semi-automatically segmented to delineate between adipose and lean muscle tissues. Two important findings were reached utilizing this data. First, a clear distinction between the musculature of males and females was established that provided 100% accuracy in being able to predict gender. Second, degenerative lumbar spines were accurately predicted at a rate of up to 92% accuracy. These results solidify prior assumptions made regarding sexual dimorphic anatomy and the pathogenic nature of degenerative spine disease.

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List of Tables

Table 1: Example of four frequently utilized filtering kernels and a version that includes the bias adaptive modification proposed in this dissertation. ... 37 Table 2: Notation for orders of operation. ... 42 Table 3: Average performance of the four filters on the synthetic phantom and Lena with various levels of

noise. Statistically significant (p<0.01) performance values for a matched pairs t-test are indicated with the appropriate symbol (footnote). Bolded values indicate the filter that obtained the best average performance and that a test of significance was performed (see footnotes). ... 50 Table 4: Average (N=10) performances of the non-local means (NLM), anisotropic diffusion (AD), and

Gaussian (G) filtering, and the bias adaptive (ba) counterparts on the Apollonian test phantom. Statistically significant (p<0.01) performance values for a matched pairs t-test are indicated with the appropriate symbol (footnote). Bolded values indicate the filter that obtained the best average performance and that a test of significance was performed (see footnotes). ... 51 Table 5: Population demographics and statistics ... 53 Table 6: Classification accuracy percentages of test Images 1 and 2 (Figure 5.3). Reported are the percentages

of pixels classified as: true positive (TP), true negative (TN), false positive (FP), false negative (FN), and percent accuracy ... 57

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Table 7: Dice values for the mean differences over twenty-six subjects at each vertebral level and muscle group for baNLM against NLM ... 58 Table 8: Average volumetric (cm3) data across the study population of three bilateral lumbar muscle groups

shows a significantly different cross-gender comparison in all lean muscle groups for a Kruskal-Wallis test. In the adipose tissue, only the multifidus showed significant differences. ... 67 Table 9: Compression rations of the intervertebral spacing from the supine-to-standing positions. Bolded

values indicate significantly different values for 𝜶 = 𝟎. 𝟎𝟏. ... 81

Table 10: Means, standard deviations, and corresponding p-values of the intervertebral spacing for the healthy plus LBP groups versus the degenerative group. Bolded values indicate significant differences with a Kruskal-Wallis test. ... 81 Table 11: Prediction accuracy of all 17-features used to classify the population. ... 84 Table 12: Prediction accuracy of a single feature (L1–L5 distance) used to classify the population. ... 86 Table 13: Maximum classification accuracy using the optimized features of the reduced (5-feature) space. ... 88

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List of Figures

Figure 1.1: Simplified drawings of the spinal column showing the cervical thoracic, lumbar and pelvic regions of the spine (right). Annotated are the lumbar vertebrae, spinal curves, and intervertebral discs (left). .. 3 Figure 1.2: Simplified sketch of L3 highlighting relevant anatomy of the vertebrae. ... 4 Figure 1.3: Sample MR slice at L4 with bulk muscle boundaries highlighted. The highlighted muscles include

the bilaterally occurring: psoas (red), erector spinae (green), and multifidi (blue). (Note: the image has been brightened and contrasted to enhance relevant tissue boundaries. ... 7 Figure 4.1: MR images of the lumbar characteristically exhibit globally inhomogeneous signals with high

SNR values near the posterior midline of the lumbar (a). The signal dissipates radially from that point along a smooth gradient, the extracted bias images resembles that of vignetting distortion observed in most optical system (b). A graph of the intensity in the out-of-plane axis (z-axis) reveals the distorted nature of spatially sensitive signal intensities (c). ... 25 Figure 4.2: A realistic bias field (a) is corrupted with 9% Rician noise (b), and then corrected for the intensity

inhomogeneity with LEMS (c). Finally, local SNR values are calculated (d). As expected, the SNR image follows a similar trend to the intensity of the extracted bias field (a). ... 31

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Figure 4.3: Three similarity windows aligned at various coordinates along the bias image (right). The mean pixel intensity contained in each window reflects the probability density (left) for parameterizing the adaptive kernel function. ... 33 Figure 4.4: The phantom and Lena (a, g) corrupted with a realistic inhomogeneity bias field (b, h) and 9%

Rician noise (c, i). A zoomed in section of the original images (d, j), with the added field (e, k), and 9% Rician noise (f, l). ... 40 Figure 4.5: Two paths utilized for the analysis of baNLM an example MRI. The method for 𝚷𝜷𝝀 (top, red)

employs LEMS prior to filtering, while the method for 𝚷𝝀𝜷 (bottom, green) employs LEMS before and

after filtering. ... 44 Figure 4.6: The mean values over 10 iterations of the optimized image performance measures, FOM, SSIM,

and 𝒊𝐑𝐌𝐒𝐄, at the various indicated noise levels for 𝚷𝝀𝜼 and 𝚷𝜼𝝀 for the synthetic phantom and Lena

(a and c, respectively). The filer strength parameter, 𝒉, that achieved the highest mean metric

measurements over 10 iterations provided an indication of the optimal value at the given noise levels for the synthetic phantom and Lena (a and c, respectively). ... 49 Figure 5.1: Sample MR slice at L4 with bulk muscle boundaries highlighted. The bilaterally occurring

muscles are the psoas (red), erector spinae (green), and multifidi (blue). Note that the image has been brightened and the contrast increased to enhance the tissue boundaries. ... 54 Figure 5.2: The original MR image (a) and compliment of the extracted bias field (e). Shown are the unbiased

image (b) and the unbiased image with bias adaptive NLM (f). Zoomed regions of the unbiased image are shown in (c) and (d), while (g) and (h) are the same zoomed regions for the bias guided filtering. Greater filtering was applied at (g) compared to (h), while little filtering occurred at (d) compared to (h). ... 58 Figure 5.3: Automatic segmentation accuracy of muscle tissue for two different MR images (a, b). Each image

was processed independently with baNLM and NLM, and then classified with fuzzy c-means clustering. The classification results of the images from both filters are overlayed on a single image and color coded

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were mutually correct, while blue indicates where they were mutually incorrect (b, d). Red indicates baNLM was correct and NLM was incorrect; while yellow indicates NLM was correct and baNLM was incorrect (a, c). Incorrectly classified pixels were identified as both false-positives and false-negatives, while correctly identified pixels were associated with only true-positive muscle tissue. The classifications from both images show baNLM (yellow) had significantly less mutually exclusive misclassifications than NLM (red) in both images (c). ... 59 Figure 5.4: Dice values for the mean differences over twenty-six subjects at each vertebral level and muscle

group for baNLM against NLM. ... 60 Figure 5.5: Population average of the paraspinal muscle volumes at specific lumbar vertebral levels. ... 63 Figure 5.6: Gender specific relationships between percent adipose tissue and BMI. Percent adipose is a

reflection of the content across the entire paraspinal column of the lumbar. Female subjects have positive correlation of adipose to BMI while males exhibited a negative correlation. ... 64 Figure 5.7: Observed gender differences in the muscle and adipose volume. Muscle volume was statistically

less for females than males (left), while no significant difference was observed in the adipose volume (right). ... 65 Figure 5.8: Boxplot of the BMI separated by gender. No significant differences were observed between

genders. ... 66 Figure 5.9: Three dimensional volumetric reconstruction of the lumbar spine and paraspinal musculature of

a female (top) and male (bottom). The surface models show a significant gender differences in both the muscle girth and adipose tissue content. ... 68 Figure 5.10: Box plots depicting gender specific percent lean muscles values. Male individuals clearly have

more lean paraspinal muscles than females. ... 69 Figure 5.11: Between gender percentages of adipose tissue in the paraspinal muscles of L1–L5. Female

percentages were significantly greater than that of males and showed a slightly less steep regression line. ... 71

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Figure 5.12: Combined bilateral (left and right) percent adipose tissue versus BMI with corresponding regression correlation lines for the psoas (a), multifidus (b), and erector spinae (c) muscles. ... 72 Figure 6.1: Distribution of lean muscle values across the three subject groups. Distributions and means are

not significantly different from each other. ... 76 Figure 6.2: Boundaries of the bones are segmented in slices of a CT scan. Adjacent segmentations are joined

together to generate 3D model of the vertebrae. ... 77 Figure 6.3: Overlay of 3D models on fluoroscopic imaging allowing for in vivo tracking of the vertebrae with

six degree of freedom resolution during a flexion-extension activity (a). Overlay with separately colored surface models are tracked to determine the relative intervertebral kinematics during a lateral bending activity (b). ... 79 Figure 6.4: Hermine spline connecting the vertebral centroids by their 3D surface models. ... 80 Figure 6.5: The Curse of Dimensionality suggests that there exists an optimal number of features for

classification; however, after exceeding this arbitrarily optimal number the performance experiences a sharp decline in accuracy. ... 85 Figure 6.6: Variance of individual principal components (bar) the cumulative sum of the first eight principal

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Acronyms and Abbreviations

NIH: National Institutes of Health

US: United States

LBP: Low Back Pain

FSU: Functional Spine Unit

MRI: Magnetic Resonance Imaging

MR: Magnetic Resonance

OMT: Osteopathic Manipulative Therapy

CT: Computed Tomography

WLS: Weighted Least Squares

SNR: Signal-to-Noise Ratio

IIC: Intensity Inhomogeneity Correction

RF: Radio Frequency

LEMS: Local Entropy Minimization with Bicubic Splines

NC: Noise Correction

DFT: Discrete Fourier Transform

NLM: Non-Local Means

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RMSE: Root Mean Squared Error

FOM: Figure of Merit

SSIM: Structural Similarity Index

MMV: Mean Metric Value

BMI: Body Mass Index

ANOVA: Analysis Of Variance

FCM: Fuzz C-Means

CSA: Cross-Sectional Area

TP: True Positive

TN: True Negative

FP: False Positive

FN: False Negative

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Chapter 1 Background

According to the National Institutes of Health (NIH), approximately a quarter of the population in the US will experience at least one day of low back pain (LBP) during a 3-month span. Other estimates indicate that: 80-90% of people in the United States will experience LBP at some point in their lifetime [1-5]; up to 62% of people will report an episode within one to two years [3, 6, 7]; it is the third most common reasons for doctor visits [8]; it is estimated that chronic low back pain accounts for 33% of the cost of all workers’ compensation claims [9]; and in terms of treatment costs and work days lost, the fiscal impact amounts to approximately $100 to $200 billion annually [10]. Yet, despite these burdens, the etiology and pathogenesis of LBP is largely unknown, as more than 85% of low back pain complaints are diagnosed as non-specific with unknown origins [11-13].

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of the trunk. Along the spinal column, the lumbar region is comprised of five vertebrae—L1 through L5—which provide structural stability and mobility to the trunk (Figure 1.1). It is an exceedingly complex system of soft and boney tissues that function synergistically to propel the torso about the lower extremity.

Located between adjoining vertebrae are intervertebral discs that provide cushioning and support for the spine. The discs form a fibrocartilaginous joint that: (1) allow for modest six degree-of-freedom motion and (2) hold the vertebral bodies together with tensile forces that function similar to a ligament. In addition, the lumbar vertebrae are also joined by two sets of bilateral synovial joints at the inferior and superior facets (Figure 1.2). The facet joints provide structural support and limit the range-of-motion between adjacent vertebrae. This “articular triad” of two facet joints and the intervertebral disc form a functional spine unit (FSU). Although motion at a single FSU is relatively limited, articulation across multiple FSUs stacked on top of each other allow for considerable global motion.

The sacrum and coccyx are located at the base of the vertebral column (Figure 1.1), and are comprised of fused, rudimentary vertebrae. The sacrum contains five vertebrae (S1 through S5) that are generally fused by early adulthood. Inferior to the sacrum, the coccyx is comprised of between three to five vertebral analogs. Gray indicates that the bones of the coccyx are likewise joined by adulthood [14]; however, other more recent studies suggest that the coccyx is more likely to be constructed of two to three separate segments [15, 16]. Nonetheless, neither the sacrum nor the coccyx contributes motion to the trunk or lumbar.

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Figure 1.1: Simplified drawings of the spinal column showing the cervical thoracic, lumbar and pelvic regions of the spine (right). Annotated are the lumbar vertebrae, spinal curves, and intervertebral discs (left).

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Figure 1.2: Simplified sketch of L3 highlighting relevant anatomy of the vertebrae.

This leaves the lumbosacral joint (L5-S1) as the biomechanical axis between the sacrum and trunk. Thus, it is common practice in kinematic analysis to represent motion of the lumbar vertebrae with a six degrees of freedom (three translational and three rotational) relative to the sacrum.

Another feature that is significant to the biomechanical function of the lumbar spine is the lordotic curve of the spine (Figure 1.1). As opposed to the kyphotic curve observed in the thoracic spine (Figure 1.1), lordosis is inwardly anterior curvature of the spine and is a biomechanically unfavorable configuration due to increased asymmetrical loading in the FSU. That is to say that the downward gravitational force is displaced onto the facet joints, while

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forces are offloaded from the dampening properties of the intervertebral disc. This notion is made further apparent by epidemiological evidence showing significantly higher incidence of back pain at the lumbar and cervical levels compared to the thoracic [17]. These naturally occurring anatomical conditions leave the lumbar (and cervical) spine inherently more susceptible to injury.

In addition to the boney anatomy, the soft tissue, chiefly the musculature, that surrounds the lumbar vertebrae play a crucial role in the biomechanic function of the lumbar spine. The lumbar back muscles, often referred to as the paraspinal muscles, add structural stability to the spine and also act directly on the vertebrae themselves to assist in propelling the trunk. From the perspective of gross anatomy, muscles of the lumbar can be classified into three distinct groups: 1) psoas major, 2) quadratus lumborum and lateral intertransversarii, and 3) interspinales, intertransversaii medials, multifidi, erector spinae (longissimus and iliocostalis) [18]. However, strict anatomical classification of the lumbar paraspinal muscles becomes problematic when intermuscular fascial boundaries are not clearly delineated through medical imaging. This is often the case in MRI studies where poor superoinferior resolution distorts between slice interpolation of the shorter and less clearly defined muscles such as the medial and lateral intertransversarii, and the boundary between the longissimus and iliocostalis. To mitigate misclassifications, more encompassing bulk boundary definitions have been devised that group muscles not only by function, but also by proximity as well.

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Strategies for evaluating bulk lumbar musculature via MRI centers around devising boundary definitions that can be clearly and consistently segmented across any given population set (Figure 1.3). Bulk analysis of lumbar MRIs reveals that three distinct bilateral muscle groups emerge which are easily identifiable across a large population base [14], these are the:

(1) Psoas Major (Figure 1.3, red): long spanning muscles that originate at the transverse

processes of T12 through L5 and insert at the lesser trochanter of the femur. It lies anterior to the vertebral bodies and slightly lateral of midline. Its primary actions are to flex the hip joint and to bend the trunk forward (when acting bilaterally from the point of insertion) [19]. Any secondary action(s) of the psoas major is not well understood and has incited significantly more controversy [20-34]. Hansen et al [18] provide a concise elucidation of the numerous hypotheses that have been proposed, which have more or less reached a consensus “that the psoas major probably functions as a stabilizer of the lumbar lordosis in upright posture.” As for imaging, axial cross-sections of the psoas major show no major intersection between it and other prominent anatomical features and, thus, are generally segregated from nearby tissues with little ambiguity and high accuracy.

(2) Multifidi (Figure 1.3, blue): part of larger system of muscles known as the

transversospinales. The muscles that are contained within the lumbar region of the transversospinal system include the: multifidi, rotatores, interspinales, and intertransversarii. The multifidus is the most prominent and longest spanning of the

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Figure 1.3: Sample MR slice at L4 with bulk muscle boundaries highlighted. The highlighted muscles include the bilaterally occurring: psoas (red), erector spinae (green), and multifidi (blue). (Note: the image has been brightened and contrasted to enhance relevant tissue boundaries.

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transverospinales, while the other three muscles are in general short and only span 1-3 vertebrae. The multifidi are situated in the lamina groove formed by the large transverse and spinous process where the form fascicles that originate at the mammillary processes of inferior vertebrae and insert to the spinous process of the vertebrae one-three vertebrae above. In addition, both the rotatores and the interspinales have insertion points along the spinous processes of the lumbar. Accurate differentiation of these three muscles proximal to insertion is difficult due to insufficient resolution or clear definition of boundary lines. However, what are clearly delineated are the lateral boundaries of the erector spine and the anterior and medial borders of the boney anatomy of the transverse and spinous processes. Since it can be difficult to differentiate the three muscles, it is common practice for researchers to simply label them collectively as the multifidus. This

technique reduces misclassifications, while also simplifying biomechanical modeling and analysis. The last muscle of the transversospinal system in the lumbar region, the

intransversarii, is a short muscle occupying the space between the transverse processes of adjacent vertebrae (laterales) and also the space from the accessory process of a vertebra to the mammillary process of the vertebra below (mediales). Gray proposed that the function of the multifidi were to assist the vertebrae in extension, lateral flexion, and rotation [14]. Later studies have indicated that when viewed as singular muscles their role appears to produce stabilizing forces rather than function as the primary drivers of motion [35]. Further studies were conducted via electromyography which supported that the multifidi’s primary function is to control intersegmental motion [36-38].

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(3) Erector Spinae (Figure 1.3, green): a large muscle that encompasses the cervical,

thoracic, and lumbar regions of the spine [14]. Somewhere along the upper lumbar region the muscle splits into three columns: a lateral (iliocostalis), an intermediate (longissimus) and a medial (spinalis). In the lumbar bulk regions of the erector spinae lie in the groove formed by the transverse and spinous processes, lateral to the multifidus. This is where they are additionally divided into the superficial (longissimus) and the deep (iliocostalis) layers. Specific location of branching of the erector spinae is largely ambiguous between subjects in a given population. Thus, for comparative analysis it is more robust to simply classify the corresponding muscular and tendinous regions by a single, clearly defined anatomical feature (i.e. the outer fascial boundary). Unilaterally flexion of the erector spinae bends the spine laterally, while bilateral flexion generates posterior sagittal rotation. In this respect, they supplement the multifidi by resisting flexion caused by the abdominal muscles during rotation of the trunk [39].

It is important to note the absence of the quadratus lumborum in the description of the transversospinal system provided above. The quadratus lumborum is not used in the analysis provided by this manuscript; still it remains a significant contributor to the functionality of the lumbar [14, 18]. The quadratus lumborum originates at the iliac crest and inserts to the apices of the transverse processes of the first four lumbar vertebrae [14]. It has been suggested that it assists with lateral flexion of the upper four lumbar vertebrae [40, 41].

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A wide range of diagnoses reveal themselves during diagnosis of LBP. It turns out that some discernable etiologies stem from mechanical changes such as deformities in the boney anatomy or visible degeneration of the intervertebral discs. Imaging studies can quickly reveal definitive diagnoses for which unambiguous algorithms are applied for treatment. However, what is perhaps of greater interest are patients that present with non-specific low-back pain that lack any definitive diagnostic evidence. In these situations, the most likely culprit could be soft tissue injuries or anomalies that do not present during imaging.

Low back pain is distinguished broadly as being either acute or chronic – depending on the length of time that pain has persisted. Acute back pain is defined as not lasting for longer than 4 weeks, while sub-acute back pain is defined as lasting between 4–12 weeks. Acute low back pain with sudden onset is the most common and often stems from a single traumatic event such as a fall or improperly lifting a large load. Treatment of acute back injury is most often approached with conservative measures and may include: over-the-counter analgesics or non-steroidal anti-inflammatory drugs (NSAIDs), heat, limited exercise, time, or osteopathic manipulation. However, though a majority of low back pain will resolve without further intervention, approximately 20% of acute injuries progress beyond the sub-acute phase and into the chronic period, with symptoms often persisting for over a year after onset.

Chronic back pain is defined by persistence for 12 weeks or longer. Similar to acute injury, chronic pain can be initiated with a single traumatic event; however, chronic symptoms often stem from structural defects that take years, if not decades, to present. Therefore, the

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pathogenesis of chronic pain is mechanical in nature and will require some sort of interventional therapy. Since a plethora of sources are responsible for the chronicity of low back pain, it is perhaps clearer to present it as a function of treatment rather than diagnosis. Depending on the post-acute diagnosis, treatments can be broadly grouped into two types of treatments:

(1) Non-surgical therapies are performed during the sub-acute period and are preferred to their counterpart, as non-surgical methods are: cheaper, less invasive, and have shown significant efficacy for specific diagnoses. Some common non-surgical treatments include: physical therapy and OMT, where improved joint flexibility and muscle strength are sought; narcotic and non-narcotic analgesics, NSAIDs, and muscle relaxants; traction, where the vertebrae are pulled slightly apart to alleviate compression of the intervertebral discs; behavioral modification; and, lastly, epidural steroid injections at the nerve root, which are the most invasive of the non-surgical treatments. Often 2-dimensional (2D) radiographic studies provide better insight and can be used to rule out any specific systemic etiologies. In terms of overall patient care, clinicians may utilize a shotgun approach that will include a combination of any of these therapies. Finally, upon successful remediation of pain, treatment is halted.

(2) Surgical therapy remains as the final and most drastic option for alleviating chronic back pain. Some common diagnoses that elicit surgical intervention include: herniated disc, spinal stenosis, spondylolisthesis, vertebral fractures, and discogenic disease. In addition to the radiographs captured during the sub-acute period, pre-operative imaging includes

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either computed tomography (CT) or magnetic resonance imaging (MRI) studies. The imaging modality utilized is specific to the diagnosis and may even be patient specific (e.g. metallic arthroplasty hardware or implanted devices may prevent use of an MRI). In addition, intraoperative imaging, such as real-time fluoroscopy, is also an option.

One unfortunate aspect to a shotgun approach is that the specific etiology is never determined and the incidence is quite simply classified as “non-specific”.

Though the occurrence and fiscal impact of LBP has been well documented in the literature, no clear consensus has been reached on the pathogenesis or etiology. This is particularly evident with nonspecific LBP where diagnosis, treatment, and prognosis can be highly variable. Ultimately this lack of a clear medical outcome is a result of a lack of understanding behind the complex nature of LBP and progressive nature of degenerative spine diseases.

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Chapter 2 Literature Review

Drawing correlation from soft tissue data of the lumbar spine to any specified pathology first requires accurate segmentation of anatomically relevant features. Through the use of sophisticated medical image processing algorithms this information is readily available in the form of MR images, but from the perspective of implementation, accessing that data is problematic due to inhomogeneous signal intensity inherent to MRI technology. Developing a reliable algorithm for accurately segmenting this data requires a thorough formulation of the noise models that confound MRI of the lumbar spine.

2.1 Noisy MRI Data

A common form of corruption encountered in magnetic resonance imaging (MRI) of the lumbar spine is spatial inhomogeneity of the signal intensity, or intensity inhomogeneity. Intensity inhomogeneity presents difficult obstacles for many modern noise filters, primarily due to an

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assumption of a globally homogenous signal-to-noise ratio (SNR). In the presence of local intensity perturbations, static filtering can lead to over or under filtering that spatially mimics the local shifts in SNR. Therefore, when confronted with intensity inhomogeneity it may often be more desirable to utilize adaptive filtering that is sensitive to local changes in signal strength.

2.1.1 Intensity Inhomogeneity Correction (IIC)

The development of IIC methods is not a trivial task and has been the principle focus of researchers for decades [42-44]. Though a number of sources have been attributed to the spatial intensity degradation of MR images such as inappropriate coil tuning, gradient eddy currents, radio frequency (RF) standing wave effects, and RF penetration effects – the predominant source of corruption has generally been linked to spatial inhomogeneity sensitivity of the RF receiver coils [45]. Intensity inhomogeneity manifests when the receiver coil suffers from a steep decline in sensitivity in the direction of increasing tissue depth.

A plethora of IIC methods have been proposed in the literature, [46] provides a review and classification of these methods. Two IIC philosophies that have garnered recent attention are: information minimization, [47, 48]; and histogram matching [49, 50]. A hybridization technique was proposed by Salvado et al. [51], which merges ideas from information minimization and histogram matching to form what they term Local Entropy Minimization with bicubic Splines (LEMS). LEMS was utilized in the ideas proposed in this dissertation and more detailed description of the algorithm in Section 2.1.

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2.1.2 Noise Correction (NC)

The general consensus behind electronic noise is that it arises during the reformulation process of the raw analog signal. MR images are most commonly reconstructed from their raw analog signals by computing the inverse discrete Fourier transform (DFT) of the k-space [52-54]. Because of difficult to control phase errors in the acquisition process, the signal measurements contains both real and imaginary parts of the DFT. Therefore, it has been shown that the noise component of the raw image signal arises from two independent Gaussian random variables (originating from the real and imaginary channels) [52-54]. By definition, the noise originating from two independent Gaussian random variables has Rician distribution.

Similar to IIC, a multitude of methods and philosophies have been developed over the years for MR NC. A review by Milanfar [55] provides a clear elucidation of the similarities that exist between most modern image filters. The author points out that the distinction is found merely in the specific formulation of the kernel function, while all other parameters and function remain more or less the same. In this regard, the kernel function is essentially an evaluation of pixels between their: spatial distance, photometric distance, or spatial and photometric distance. Our analysis focuses on the non-local means (NLM) filter, which uses photometric distances to develop the kernel [56, 57]. However, the selection of NLM was mostly arbitrary and the analysis could theoretically be extended to any of the kernels listed above. A more detailed discussion of the NLM filter is presented in Section 4.1.2.

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The development of the IIC and NC methods were by in large designed as independent entities tailored to one specific artifact or the other. However, the crux of our analysis focuses on the situation when confronted with both artifacts in the same image, and, more specifically, how to perform filtering with global spatial intensity non-uniformity. Various techniques have been proposed in the literature to combat such a scenario, such as coupling a filtering parameter calculated from local noise statistics with an anisotropic filter [58]. Another study developed a variational approach that utilized a half-quadratic optimization [59] NC method by adaptively fitting the extracted bias field through simultaneous body and surface coil images [60]. Kervrann and Boulanger [61] developed a spatially adaptive filter, similar in structure to the NLM filter, that optimizes the weighting function by iteratively growing the search window and adaptively updating the filtering parameter.

In developing an algorithm that encompasses both IIC and NC it is also important to consider the order in which the corrections methods are applied. That is, is it more optimal to perform IIC first and then apply NC, or is the inverse procedure more optimal? Previous studies have concluded that IIC should precede NC, since IIC can disrupt the homogeneity of the embedded image noise [62, 63]. However, it is worth noting that these studies employed denoising methods with static filter strength parameters and, thus, did not investigate the relevance of adaptive noise filters in the image processing sequences.

A common measurement model for describing the correlation between the observed noisy image and the latent signal of the true image:

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𝑦(𝑖) = 𝑥(𝑖)𝑏(𝑖) + 𝑛(𝑖) (1)

where, at the 𝑖-th pixel, 𝑦(𝑖) is the observed signal, 𝑥(𝑖) is the signal of interest, 𝑏(𝑖) is the intensity inhomogeneity, and 𝑛(𝑖) is noise.

Solving for (10) elicits the question of which correction method should be performed first to achieve the most optimal results. Previous studies have concluded that IIC should precede NC, since IIC can disrupt the homogeneity of the embedded image noise [63, 64]. However, these studies employed static denoising methods and, thus, did not investigate the relevance of adaptive filtering.

The aims of this study were twofold: first was to modify a non-local means filter to adaptively update its kernel weight with values based on estimates of the underlying bias noise. Using this technique, the strength of filtering is adaptively modulated by local SNR values, where low SNR regions experience more filtering and high SNR regions incur less filtering. The second aim was to determine the order in which LEMS and adaptive NLM should be performed. We evaluated the performance of both aims with a synthetic phantom and MR images of the lumbar spine.

2.2 The Kernel

A filtering kernel, sometimes referred to as a convolution matrix, is a useful tool frequently employed in image processing for smoothing, blurring, sharpening, and embossing, to name a few. It is a small matrix that is convolved (Appendix A) with the original image to achieve the

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desired effect, such as a filtering (i.e. smoothing). Returning to Equation (1) and for the purposes of the formulizing the smoothing kernel, the bias signal, 𝑏(𝑖), and the latent signal of interest, 𝑥(𝑖) are combined by elementwise multiplication such that 𝑥(𝑖)𝑏(𝑖) = 𝑧(𝑖). The algorithm most frequently employed for solving for the true signal of interest is the weighted least squares (WLS) approximation. Adopting the notation used by Milanfar [55], the WLS approximation of the true signal 𝑧(𝑣_𝑗) at position 𝑣 is defined as:

𝑧̂(𝑣𝑗) = arg min𝑧(𝑣𝑗)∑ [𝑦𝑖 − 𝑧(𝑣𝑗)]

2

𝐾(𝑣𝑖, 𝑣𝑗, 𝑦𝑖, 𝑦𝑗) 𝑛

𝑖=1 . (2)

where 𝑦_𝑖 − 𝑧(𝑣_𝑗) is the error and 𝐾(∙) is the kernel, or weighting function, defined with respect to the indices 𝑖 and 𝑗. 𝐾(∙) Is a symmetric, unimodal function that measures the similarity between the samples 𝑦𝑖 and 𝑦𝑗 at locations 𝑣𝑖 and 𝑣𝑗, respectively. This representation of the

WLS is extremely convenient as it demonstrates how the kernel is clearly parsed from the error expression so as to simplify and promote a discussion of how the kernel affects the performance of the filter. Milanfar goes on to construct four generalized types of filters that differ only in the design of the kernel, which utilize the spatial and/or photometric distances [55]. Here, three of those kernel models are modified and juxtaposed against the adaptive counterparts proposed later in this dissertation (Table 1).

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Chapter 3 Research Aims

The primary goal of this dissertation was to investigate the etiologic and pathogenic characteristics of LBP, in addition to searching for possible links that would expose LBP as a progressive disease that eventually degrades overtime leading to a host of more serious diagnoses and treatments. In order to determine a causal link between LBP and progressive degeneration, spines with degenerative disease were also investigated in this work.

The best and most accurate in vivo method for drawing etiological inference of soft tissue involves extracting information via clinical imaging. Numerous hypotheses suggest a possible correlation between musculature and adipose tissue of the lumbar spine [65-69]. Differentiation between these tissues is clearly visible in T2 MRIs where they appear with deeply contrasted

edges, as muscle appears dark, while adipose is nearly white. However, one significant drawback to T2 MRI of the lumbar spine is the presence of a globally inhomogeneous signal-to-noise ratio

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(SNR). This results in deleterious distortion of image quality, which in turn severely hinders the fidelity of intensity based segmentation algorithms.

One noteworthy aspect of segmentation in medical imaging is that there is generally a preprocessing step that involves smoothing electronic noise which significantly boosts the performance accuracy. This does not pose any hindrance in most applications of medical imaging. However, a serious problem is invoked when an image is confounded with both intensity inhomogeneity and electronic noise. Neither IIC nor NC methods were originally designed to compensate for both forms of distortion. Therefore, it is the primary aim of this dissertation to propose, test, and validate a method for reliably smoothing electronic noise when in the presence of global intensity inhomogeneity. Whereby developing an algorithm that successfully merges the two concepts (IIC and NC) into a more singular image processing technique. Namely, the design of a filter that utilizes information gained from IIC process and utilizes it to enhance filtering for NC.

The proposed filter was designed with adaptive properties that determine local SNR values based on the underlying intensity bias and adjusts the strength of filtering accordingly. The framework of this dissertation is built around the non-local means (NLM) filter, a smoothing technique that is popular in the medical imaging community. Thusly, the technique proposed in this manuscript is termed bias adaptive non-local means (baNLM) filtering. The efficacy of baNLM is measured against traditional NLM filtering by way of testing with a synthetic phantom, a common black-and-white photograph (Lena), and T2 MRIs of the lumbar spine.

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Another primary aim of this dissertation was to elucidate any etiologic and/or pathogenic information of LBP and spine degeneration within the study population. As was previously stated, the objective was not to make sweeping generalization about LBP, but rather to take a small sample size, acquire copious amounts of data on each subject, and attempt to expound correlations that in the future will provide some clarity with where to look and what to look for in larger population based studies. Because, as it currently stands, neither where nor what have been explicitly defined with any degree of certainty that would be acceptable in the clinical setting.

A secondary, and perhaps coincidental, objective of this dissertation was to investigate differences that exist between male and female lumbar spines. Though sexual dimorphisms are not radically profound in the world of biomechanics, they do represent a particularly noticeable exemption in the literature surrounding the spine. While not wholly devoid of research, any conclusions that may have been drawn are insufficient for in the clinical setting [19, 70-74].

Finally, it would be a mistake to assume that the scope of this dissertation alone could be extended to the diagnostic algorithms currently employed by clinicians, or that it might draw sweeping conclusions regarding some “grand cure” to low back pain. Such an undertaking would indeed require extensive studies with a wide population base and shrewd analytical techniques (the type reserved for most of the serious scientific and clinical enquires out there that use legitimate population statistics to posit generalized inferences). Rather, the breadth and scope of this dissertation aims to utilize a smaller sample size, with which a wealth of data was collected,

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and distill out clinically relevant findings as they pertain to soft tissues of the lumbar. This is ultimately done with the hope that the ideas developed here may later be extended to someday enhance treatments and clinical outcomes of lumbar spine related disease.

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Chapter 4 Adaptive Filtering

As was demonstrated in Chapter 2, MR images of the lumbar spine quite often suffer from a globally inhomogeneous signal-to-noise ratio (SNR) throughout the field of view, which can severely obstruct the robustness of modern post-processing imaging techniques often implemented for filtering and segmentation. The primary premise of the research presented in this manuscript hinges on the accuracy of segmenting muscle data from MRIs, thus it becomes of vital importance to develop a robust filtering algorithm for medical images that can aptly manage globally inhomogeneous SNRs that inherently hinder intensity based methods for segmentation. This chapter focuses on the theory, implementation, and testing of the development processes behind designing an image filter that can adaptively and optimally smooth regions based on their local SNR values.

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4.1 Understanding the Noise Model

T2 MRI of the lumbar spine notoriously suffers from a shift in intensity that tends to decrease

laterally and anteriorly from most posterior midline of the back (Figure 4.1(a, b)). As previously mentioned, this shift is due to a one sided coil that is situated between the subject and the scanning table. The purpose of the coil (or any MR coil for that matter) is to amplify tissue signals to make for clearer reading by clinicians, yet since the lumbar coil is only one-sided, the quality of the signal quickly declines the further away the tissue is from the coil. It is worth noting that coils used in, for example, brain and knee scans completely encompass the anatomical region of interest and thus avoid problems associated with inhomogeneous signal strengths. Nonetheless, the typical lumbar MR scan will quite reliably have high SNR the nearest the most posterior midline aspect of the lumbar, by the spinous processes, and will quickly fade moving anteriorly and laterally (Figure 4.1(b)).

The effect of inhomogeneity severely distorts relative intensity values between tissue types. For example, the image used in Figure 4.1(c) shows fatty tissue captured in the posterior spine (labeled “High SNR”) with near maximum pixel values of around 255, whereas tissue captured anterior to the spinal column (labeled “Low SNR”) show fatty tissue with pixel values around 20. Ideally these fatty tissues would have similar pixel values to make it easier to distinguish from other tissue types. Alas, it is understandable why this does not pose a significant problem in the clinical setting as the human brain is readily capable of interpolating variable tissue signals by recognizing that though their intensity may appear different in different locations, the tissues in both regions are in fact the same, the underlying signal strength has simply undergone some

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Figure 4.1: MR images of the lumbar characteristically exhibit globally inhomogeneous signals with high SNR values near the posterior midline of the lumbar (a). The signal dissipates radially from that point along a smooth gradient, the extracted bias images resembles that of vignetting distortion observed in most optical system (b). A graph of the intensity in the out-of-plane axis (z-axis) reveals the distorted nature of spatially sensitive signal intensities (c).

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degradation along a gradient. It turns out that through visual inspection a clinician is able to quickly and efficiently segment relevant tissue boundaries to conclude a diagnoses. However, the unfortunate realization quickly settles in that intensity based segmentation algorithms are not nearly as clever as the human brain.

Formulizing the noise model associated with MR imaging is necessary for understanding how best to approach image reconstruction when faced with this sort of distortion. The first step is to investigate the nature of MR based inhomogeneous signals and how that correlates with relative SNR values. The discussion begins with a brief introduction on the method that was utilized in this research for intensity inhomogeneity correction (IIC). As was described in Chapter 2, numerous methods for correcting images with globally inhomogeneous signals have been proposed in the literature; however, LEMS has shown promise in the domain of MR imaging and also provides a practical framework from which to build upon.

4.1.1 Local Entropy Minimization by Bicubic Splines (LEMS)

LEMS is an optimization algorithm that determines the global background bias field through noise estimates in local sub-regions of the global image. The authors posit that the entropy of an image is a relevant analogy by which to represent noisy image, and that by estimating the entropy in smaller sub-regions of the image, these entropies can be ranked to generate a piecewise representation of local SNR values. This idea is formalized by adopting the variable definitions outlined in Eq. (1) for a noisy image, where the entropy, 𝑆, is given by: 𝑆 =

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of 𝑋 and 𝒩 is a window containing a targeted local sub-region. The values of 𝑃𝐷𝐹_𝑋(𝒩) are approximated by a histogram of the pixels in the current region divided by the number of pixels in that region. LEMS optimizes the entropy in a piecewise fashion by calculating the entropy in local square neighborhoods (or knots). The knots are then ranked based on their SNR. At the first knot, the bias field is optimized by minimizing the entropy in the local region. Next, the second highest SNR knot is optimized for both the current and previous regions; the third knot is then optimized for the first, second, and third regions and so on, until all knots have been optimized. Concurrently, the local bias fields are stitched together with a bicubic spline that smooths the gradient of the global bias field. The final result is an estimation of the intensity bias the distortion observed the original image, which is also an estimation of 𝑏(𝑖) from Equation (1) and the restored image is determined by: 𝑥_{𝐿𝐸𝑀𝑆}(𝑖) = 𝑦(𝑖)/𝑏(𝑖).

At this point, the most obvious question is: why not now simply pass the undistorted image 𝑥𝐿𝐸𝑀𝑆(𝑖) through one of the filtering algorithms that is commonly found in the medical image

processing repertoire? The answer requires some insight into the theoretical basis of image filtering and the fundamental assumptions used to formulize them.

4.1.2 Non-local Means (NLM) Filtering

As was described in Chapter 2, this manuscript largely addresses kernel guided filters that utilize the weighted least squares framework outlined in Equation (2). That is not to say that other such frameworks would not be applicable in this context, the weighted least squares simply casts a wide net over many modern filters. In addition, the least squares framework also allows for one

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to readily utilize photometric and/or geometric features to populate the kernel, Milanfar discusses this implementation in greater detail [55]. Nonetheless, one popular filter that uses this framework is the non-local means (NLM) algorithm [57, 75]. The NLM filter prepares a patched non-local similarity kernel to compute the photometric distance between other local patches, while ignoring the geometric distance between those patches. Again, adopting the variables defined in Equation (1), the formulization of NLM is detailed below.

Given a noisy image 𝑦 = {𝑦(𝑖)|𝑖 ∈ 𝐼}, the true signal is estimated by a weighted average of all pixels in the image, 𝑁𝐿𝑀[𝑦](𝑖) = ∑_𝑗∈𝐼𝑤(𝑖, 𝑗)𝑦(𝑗), where {𝑤(𝑖, 𝑗)}_𝑗 represents the family of weights that depend on the similarity between pixels 𝑖 and 𝑗. This measures the grayscale similarity between the vectors 𝑦(𝒩_𝑖) and 𝑦(𝒩𝑗), where 𝒩𝑘 denotes a square neighborhood (or

the similarity window) centered at the 𝑘th pixel. The similarity is a measurement of the photometric distance, ‖𝑦(𝒩𝑖) − 𝑦(𝒩𝑗)‖_2,𝑎

2

, where 𝑎 > 0 is the standard deviation of the Gaussian kernel. The kernel is defined as:

𝑤(𝑖, 𝑗) = 1 𝑍(𝑖)𝑒 − ‖𝑦(𝒩𝑖)−𝑦(𝒩𝑗)‖_2,𝑎 2 ℎ2 _, (3)

where 𝑍(𝑖) is the normalizing constant:

𝑍(𝑖) = ∑ 𝑒− ‖𝑦(𝒩𝑖)−𝑦(𝒩𝑗)‖_2,𝑎 2 ℎ2 _. 𝑗 (4)

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The parameter ℎ dictates the strength of filtering by controlling the exponential decay of the kernel function. Larger values of ℎ flatten the decay of the weighting function, thus causing greater filtering. Smaller values of ℎ sharpen the decay, forcing the weights to zero and causing less filtering. The parameterization of ℎ is the principle point of interest in our analysis and provides the connection between image denoising and intensity inhomogeneity.

Ideally NLM compares the similarity of neighborhoods throughout the entire image. However, from the perspective of practicality such a process would be far too computationally expensive. A smaller search window is employed that encompasses the similarity. The idea is that the search window should be large enough to determine non-local neighborhoods, while still small enough to yield reasonable computation times. As was suggested by Baudes, the search window was set to be 21 x 21 and the similarity window was set to 7 x 7. These window sizes were utilized throughout the entirety of this dissertation. With the 512 x 512 MR images used in this research, the iterative complexity of the final algorithm is calculated as 5122 x 212 x 72 ~ 5.7 x 109. This brings to light one of the primary drawbacks to NLM that even in its abridged form it is still computationally impractical for many uses.

Nonetheless, one benefit to the NLM algorithm is that it does not require many external parameter definitions by the user, in fact only three. Two of which are the search and similarity window sizes. Those are not the primary focus of this analysis and so were left unchanged from what the original authors proposed. The last, which is of perhaps more interest and pertinence to this manuscript are the properties and assumptions associated with tuning the parameter ℎ.

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The assumption behind ℎ is that it will be parameterized with a value proportional to the noise level present in the image. So, an image with more noise will require an increased value of ℎ that flattens the decay of the kernel function. However, a singular value of ℎ assumes a globally homogeneous noise model throughout the image, which is certainly not the case in the present lumbar MR images and thus poses a significant predicament with regards to efficacy. This dilemma is best understood by understanding the changes that occur during the LEMS reconstruction process. Begin, for example, with a blank median image that is corrupted (by vector multiplication) with a realistic inhomogeneity bias (Figure 4.2(a)) and then 9% Rician noise is added to the image (Figure 4.2(b)). Now say the intensity inhomogeneity bias is corrected for using the LEMS method (Figure 4.2(c)). At this point, what is of particular interest is the noise model present in the post-processed image. The effects of LEMS can be observed by displaying the output image as a map of local SNR values (Figure 4.2(d)).

The goal now becomes determining how to filter the output image in Figure 4.2(c) for optimal denoising. From the image provided in Figure 4.2(d), it is quite apparent that image suffers from a wide range of SNR. Thus, if a static value of ℎ were utilized in NLM most local regions would not experience optimal smoothing and invariably undergo over- or under-filtering. Finally, we observe why the basic assumption of NLM (and almost all non-SNR adaptive filters) is inappropriate for usage with lumbar MRIs; and, further, why over/under-filtering would be far less than optimal in the context of accurate segmentation. It would rathe be significantly more desirable if a system were devised that accurately controls the decay of the kernel to reflect local

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Figure 4.2: A realistic bias field (a) is corrupted with 9% Rician noise (b), and then corrected for the intensity inhomogeneity with LEMS (c). Finally, local SNR values are calculated (d). As expected, the SNR image follows a similar trend to the intensity of the extracted bias field (a).

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runtime, and if it was also intelligently sensitive to local changes in SNR and adapt accordingly. Figure 4.2(d) shows that the SNR bears close resemblance to the original bias image observed in Figure 4.2(a). In fact, the research presented in this manuscript, posits that the bias and SNR images bear such close resemblance that the underlying bias image provides a detailed map to adaptively parameterizing NLM (i.e. ℎ) such that it accurately reflects local SNRs. From an algorithmic point of view, the system might proceed as follows: 1) the image of interest is passed to LEMS which extracts the underlying bias, 2) the bias image is passed along to some filtering scheme, and 3) the filter adaptively adjusts ℎ using the bias image as a map. The resultant would be an image with smoothed regions that correspond with local SNR values.

4.1.3 The Coupled Algorithm: Bias Adaptive Non-Local Means (baNLM) Filtering

As mentioned in Section 4.1.2, the constant ℎ in (3) and (4) is a tuning parameter that adjusts the level of smoothing imposed by the filter (lower values incur less smoothing, while higher values experience more smoothing). In various applications of NLM, ℎ is generally parameterized as a function of the global standard deviation of the noise. This is, however, impractical for lumbar MRIs where: (1) the noise variance is not explicitly known and (2) there is poor spatial homogeneity of the SNR. In the context of external inputs, the extracted bias field is directly applicable to the parameterization of NLM (Figure 4.3). This notion is illustrated by Figure 4.3 which shows the correlation between the probability density of the kernel and local SNR values. The green similarity window indicates an approximate mean value of the image and perhaps encompasses some supposed theoretical region with “average” intensity characteristics.

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Figure 4.3: Three similarity windows aligned at various coordinates along the bias image (right). The mean pixel intensity contained in each window reflects the probability density (left) for parameterizing the adaptive kernel function.

That is to say that the signal is neither too weak nor too saturating and that the noise is at some intensity that would typically be expected. For the sake of brevity, the proposed theoretical framework will suppose that NLM is tuned to this specific region. Now, if we move to the orange similarity window, which spans the brightest region of the image and likewise correlates to the highest SNR region, we could correctly assume that this region requires less filtering than that of the green. If we were to visualize the probability densities of each region, we observe that has a sharper curve, which is directly analogous to the decay of the optimal kernel (Figure 4.3 (left), orange vs. green curve). Conversely, the blue similarity window reflects a region with low SNR and where heavier smoothing would likely be desired. The noise model in that region would be represented with widen and flatten the density curve (Figure 4.3 (left), blue vs. green curve). Similar to as before, the relative sharpness of the density function is directly analogous to the optimal decay of the kernel. Thus, the blue region will require a higher value of ℎ than in the

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green region to achieve optimal smoothing.

It has now been demonstrated how the bias image can be used as a map to guide smoothing; however, one last modification must be clarified. The bias image represents the compliment (or negative) of what is required to properly augment ℎ. As previously mentioned, h effectively controls the decay of the exponential function of the kernel. Since it is the denominator of a negative exponential the function decay slows with large values, while hastening with small values. The exact opposite effect is achieved with the extracted bias image (e.g. Figure 4.2(a) and Figure 4.3(right)), thus, necessitating the inversion of the image. The compliment of the bias image is found by:

𝑏̂(𝑖) = arg max

𝐵 (𝑏(𝑖)) − 𝑏(𝑖) (5)

where 𝑏̂(𝑖) is the compliment of bias image and arg max_𝐵(∙) is the maximum gray value of the bit depth range of the bias image, 𝐵.

Maintaining the notation developed in Section 4.1.2 for the similarity window and center pixel, we define the adaptive filtering parameter as the mean of 𝑏̂ within the corresponding similarity window about the current center pixel 𝑖:

𝛽(𝑖) = 1

𝑚∑ 𝑏̂(𝒩𝑖(𝑙))

𝑚

𝑙=1

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where 𝛽(𝑖) is the adaptive weighting parameter, 𝑙 is the current pixel in the similarity window, and 𝑚 is the size of the similarity window. Consequently, 𝛽(𝑖) modulates the filtering intensity by dictating the local SNR within the similarity window. In this view, 𝛽(𝑖) is an instantaneous scalar multiplier to ℎ, which when combined with (3) is:

𝑤(𝑖, 𝑗) = 1 𝑍(𝑖)𝑒

−‖𝑦(𝒩𝑖)−𝑦(𝒩𝑗)‖2,𝑎

2

𝛽(𝑖)ℎ2 (7)

and with (4) as:

𝑍(𝑖) = ∑ 𝑒− ‖𝑦(𝒩𝑖)−𝑦(𝒩𝑗)‖_2,𝑎 2 𝛽(𝑖)ℎ2 𝑗 (8)

As shown in equations (7) and (8), the weighting values modulate the strength of filtering. In our analysis of the two test images we optimized ℎ for NLM and then initialized baNLM with the same value of ℎ to ensure an unbiased evaluation of each filters’ performance.

Adaptive kernel estimation for enhanced filtering and pattern classification of magnetic resonance imaging: novel techniques for evaluating the biomechanics and pathologic conditions of the lumbar spine

Trace: Tennessee Research and Creative

Exchange

Adaptive kernel estimation for enhanced filtering

and pattern classification of magnetic resonance

imaging: novel techniques for evaluating the

biomechanics and pathologic conditions of the

lumbar spine

Acknowledgments

Abstract

Table of Contents

List of Tables

List of Figures

Acronyms and Abbreviations

Chapter 1

Background

Chapter 2

Literature Review

Chapter 3

Research Aims

Chapter 4

Adaptive Filtering